A Stochastic Approach to Content Adaptive

Digital Image Watermarking

Sviatoslav Voloshynovskiyyz, Alexander Herrigely,Nazanin Baumgaertnery, and Thierry Punz

CUI - University of Genevaz DCT - Digital Copyright Technologiesy

Department of Computer Science Research & Development24 rue General Dufor Stau�acher-Strasse 149CH-1211 Geneva 4 CH-8004 Zurich

Switzerland Switzerland

Abstract. This paper presents a new stochastic approach which can beapplied with di�erent watermark techniques. The approach is based onthe computation of a Noise Visibility Function (NVF) that characterizesthe local image properties, identifying textured and edge regions wherethe mark should be more strongly embedded. We present precise formu-las for the NVF which enable a fast computation during the watermarkencoding and decoding process. In order to determine the optimal NVF,we �rst consider the watermark as noise. Using a classical MAP imagedenoising approach, we show how to estimate the "noise". This leads toa general formulation for a texture masking function, that allows us todetermine the optimal watermark locations and strength for the water-mark embedding stage. We examine two such NVFs, based on either anon-stationary Gaussian model of the image, or a stationary GeneralizedGaussian model. We show that the problem of the watermark estimationis equivalent to image denoising and derive content adaptive criteria. Re-sults show that watermark visibility is noticeably decreased, while at thesame time enhancing the energy of the watermark.

1 Introduction

Digital image watermarking is applied today as a popular technique for authen-tication and copyright protection of image data. Based on global informationabout the image characteristics, many approaches and commercial solutions em-bed the watermarking signal as random noise in the whole cover image with thesame strength regardless of the local properties of the image. This embeddingmay lead in practice to visible artifacts specially in the at regions which arecharacterized by small variability. In order to decrease these distortions the givenwatermark strength has to be decreased. This, however, reduces drastically therobustness of the watermark against di�erent sorts of attacks, since the imageregions which generate the most visible artifacts determine the �nal maximumstrength of the watermark signal to be embedded.

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We present for this problem an e�ective solution1 which embeds the watermarkinto the cover image according to the local properties of the image, i.e. apply-ing this technique every watermarking algorithm will be content adaptive. Incontrast to recently published results we present a new stochastic approach toidentify the regions of interest for the embedding. This approach has the advan-tage that it is applicable for very di�erent types of images and is not constrainedwith the identi�cation of an adequate set of parameters to be determined beforethe identi�cation of the local charactersitics as it is the case in [8]. In addition,the approach presented may be applied to di�erent domains, such as coordi-nate, Fourier and wavelet. We show the interrelationship to the image denoisingproblem and prove that some of the applied techniques are special cases of ourapproach. Comparing the derived stochastic models with results recently pub-lished we have noticed that some heuristically derived formula are close to ourproblem solution. The stationary Generalized Gaussian model is superior con-cerning the strength of the watermark to be embedded as well as the imagequality.

2 State-of-the-art Approaches

Some authors tried to develop content adaptive schemes on the basis of theutilization of luminance sensitivity function of the human visual sytem (HVS)[1]. Since the derived masking function is based on the estimation of the imageluminance a soley luminance based embedding is not e�cient against waveletcompression or denoising attacks. Another group of watermarking algorithmsexploites transfer modulation features of HVS in the transform domain to solvethe compromise between the robustness of the watermark and its visibility [3].This approach embeds the watermark in a predetermined middle band of fre-quencies in the Fourier domain with the same strength assuming that the imagespectra have isotropic character. This assumption leads to some visible artifactsin images specially in the at regions, because of anisotropic properties of imagespectra. A similar method using blocks in DCT (discrete cosine transform) do-main was proposed in [4]. In the context of image compression using perceptuallybased quantizers, this concept was further developed in [5] to a content adaptivescheme, where the watermark is adjusted for each DCT block. However, as theoriginal image is required to extract the watermark, the practical applications ofthis approach are very limited since it can be shown that the usage of the coverimage will results in watermark schemes which can be easily broken. AnotherDCT based algorithm which uses luminance and texture masking was developedby [6]. The next group of methods is also based on the image compression back-ground [7] and practically exploits 3 basic conclusions of the above paper: (1)all regions of high activity are highly insensitive to distortion; (2) the edges aremore sensitive to distortion than highly textured areas; (3) darker and brighterregions of the image are less sensitive to noise. The typical examples of this

1 This work has been supported by the Swiss National Science Foundation (Grant5003-45334) and the EC Jedi-Fire project (Grant 25530)

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approach are ([8], [9]). The developed methods consist of a set of empirical pro-cedures aimed to satisfy the above requirements. The computational complexityand the absence of closed form expressions for the perceptual mask complicatethe analysis of the received results. However, experiments performed in thesepapers show high robustness of these approaches. A very similar method wasproposed in [10], where edge detectors are used to overcome the problem of vis-ibility of the watermark around the edges. The goal of this paper is to developa coordinate domain content adaptive criterion which may easily be applied toany watermarking technique in coordinate, Fourier, DCT or wavelet domainsas perceptual modulation function. The basic idea of our approach consists inthe adequate stochastic modelling of the cover image. This allows the estima-tion of the image as well as the watermark and makes the application of theinformation theory to the watermarking problem possible. Knowing stochasticmodels of the watermark and the cover image, one can formulate the problem ofwatermark estimation/detection according to the classical Bayessian paradigmand estimate the capacity issue of the image watermarking scheme. Developingthis concept we show that the problem of watermark estimation is equivalent toimage denoising, and derive content adaptive criteria. Finally, we will show therelevance of the developed criterion to the known empirical results.

3 Problem Formulation

Consider the classical problem of non-adaptive watermark embedding, i.e. em-bedding the watermark regardless of the image content. In the most general caseit can be de�ned according to the next model:

y = x+ n; (1)

where y is the stego image (y 2 RN and N = M �M ), x is the cover (original)image and n is associated with the noise-like watermark image encoded accordingto the spread spectrum technique [11]. Our goal is to �nd an estimate n̂ of thewatermark n either directly, or equivalently, an estimate x̂ of the cover image xand then compute an estimation of the watermark as:

n̂ = y � x̂; (2)

where n̂ and x̂ denote the estimates of the watermark and the cover image re-spectively. The decision about the presence/absence of the watermark in a givenimage is then made by a robust detector. This detector must consider the priorstatistics of the watermark and the possible errors of its estimation due to thedecomposition residual coe�cients of the cover image and the possibly appliedattack. The generalized scheme of such an approach could be schematically rep-resented according to �gure (1). This generalized idea has found practical appli-cations in the watermarking algorithm [1] and steganography method [12]. Thekey moments of the above approach are the design of the corresponding esti-mator and the robust detector. The problem of estimation of the cover image

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from its noisy version is known as image denoising or image smoothing. We willconcentrate our consideration on its analysis and its solution in this paper, andshow a way of deriving some stochastic criteria for content adaptive watermark-ing. To solve this problem we use the Maximum a Posteriori Probability (MAP)approach.

Our stochastic approach is based on two models of the image where the imageis assumed to be a random process. We consider a stationary and non-stationaryprocess to model the cover image. The stationary process is characterized by theconstant parameters for the whole image and the non-statinary has spatiallyvarying parameters. To estimate the parameters a maximum likelihood estimateis used in the speci�ed neighbourhood set. We assume that image is either a non-statinary Gaussian process or a stationary Generalized Gaussian. In contrast tomany information theory approaches for watermarking, we don't consider thedistribution of an image as purely stationary Gaussian, since the image regionsof interest for watermarking have di�erent local features. In addition, the channelcapacity is not uniform since the image contents of every window constitute achannel capacity which is closely linked to the local image characteristics whichare not uniform over the whole image and dependent from visible artifacts.

4 Watermark Estimation Based on MAP

To integrate the watermarking problem into a statistical framework, a proba-bilistic model of the watermark and the cover image must be developed. If thewatermark has the distribution pn(n) and the cover image the distribution px(x),then according to the MAP criterion, the watermark estimate could be foundas:

n̂ = argmax~n2RNL(~njy); (3)

where L(~njy) is the log function of the a posteriori distribution:

L(~njy) = lnpx(yj~n) + lnpn(~n): (4)

The estimate n̂ , found according to equation (2), leads to the next formula-tion:

x̂ = argmax~x2RN flnpn(yj~x) + lnpx(~x)g: (5)

The problems (3), (4) and (5) are according to the formulation (2) equivalentto each other. The formulation (5) is the typical image denoising problem, andwill be considered further in the present paper.To solve this problem it is necessary to develop the accurate stochastic modelsfor the watermark pn(n) and the cover image px(x).

Under the assumption that the watermark is received using spread spectrumtechnique, it is possible to model it as Gaussian random variable. Let samplesni;j (1 � i; j � M ) be de�ned on vertices of anM �M grid, and let each sampleni;j take a value in R. Let further all the samples be independent indentically

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distributed (i.i.d) . Then:

pn(yjx) = 1q(2��2n)

N� expf� 1

2�2n(y � x)T (y � x)g; (6)

where �2n is the variance of the watermark. This assumption is reasonable, be-cause of the fact that the Gaussian distribution has the highest entropy among allother distributions and hence, from the security point of view, any spread spec-trum encoding algorithm should approach this distribution in the limit. Thus,the watermark could be modelled as n � N (0; �2n). The next important questionis the development of an adequate prior model of the cover image.

5 Stochastic Models of the Cover Image

One of the most popular stochastic image model, which has found wide appli-cation in image processing, is the Markov Radom Field (MRF) model [13]. Thedistribution of MRF's is written using a Gibbs distribution:

p(x) =1

Ze�P

c2AVc(x); (7)

where Z is a normalization constant called the partition function, Vc(�) is afunction of a local neighboring group c of points and A denotes the set of allpossible such groups or cliques.

In this paper we will consider two particular cases of this model, i.e. theGaussian and the Generalized Gaussian (GG) models. Assume that the coverimage is a random process with non-stationary mean. Then using autoregressive(AR) model notations, one can write the cover image as:

x = A � x+ " = �x+ "; (8)

where �x is the non-stationary local mean and " denotes the residual term dueto the error of estimation. The particularities of the above model depend on theassumed stochastic properties of the residual term:

" = x� �x = x� A � x = (I � A) � x = C � x; (9)

where C = I � A and I is the unitary matrix. If A is a low-pass �lter, then C

represents a high-pass �lter (decomposition operator).We use here two di�erent models for the residual term ". The �rst model is

the non-stationary (inhomogeneous) Gaussian model and the second one is thestationary (homogeneous) Generalized Gaussian (GG) model.

The choice of these two models is motivated by the fact that they have foundwide application in image restoration and denoising ([14], [15]), and that thebest wavelet compression algorithms are based on these models ([16], [17]).Their main advantage is that they take local features of the image into account.In the �rst case, this is done by introducing the non-stationary variance using a

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quadratic energy function, and in the second case, by using an energy function,which preserves the image discontinuities under stationary variance. In otherwords, in the non-stationary Gaussian model, the data is assumed to be locallyi.i.d. random �eld with a Gaussian probability density function (pdf), while inthe stationary GG model the data is assumed to be globally i.i.d..

The autocovariance function in the non-stationary case can be written as:

Rx =

0BBBB@

�2x1 0 � � � 0

0 �2x2 0...

... 0. . . 0

0 0 0 �2xN

1CCCCA ; (10)

where f�2xij1 � i � Ng are the local variances.The autocovariance function for the stationary model is equal to:

Rx =

0BBBB@

�2x 0 � � � 00 �2x 0

...... 0

. . . 00 0 0 �2x

1CCCCA ; (11)

where �2x is the global image variance.

According to equation (9) the non-stationary Gaussian model is characterizedby a distribution with autocovariance function (10):

px(x) =1

(2�)N2

� 1

j detRxj1

2

� expf�1

2(Cx)TR�1

x Cxg; (12)

where j detRxj denotes the matrix determinant, and T denotes transposition.The stationary GG model can be written as:

px(x) = ( �( )

2� ( 1 ))

N2

� 1

j detRxj1

2

� expf��( )(jCxj 2 )TR�

2

x jCxj 2 g; (13)

where �( ) =

r� ( 3

)

� ( 1 )and � (t) =

R10 e�uut�1du is the gamma function, Rx is

determined according to (11), and the parameter is called the shape param-eter . Equation (13) includes the Gaussian ( = 2) and the Laplacian ( = 1)models as special cases. For the real images the shape parameter is in the range0:3 � � 1.

Having de�ned the above models of the watermark and the cover image,we can now formulate the problem of image estimation according to the MAPapproach.

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6 Image Denoising Based on the MAP Approach

MAP estimate of the cover image results in the following general optimizationproblem:

x̂ = argmin~x2RN f 1

2�2nky � ~xk2 + �(r)g; (14)

where �(r) = [�( ) � jrj] , r = x��x�x

= Cx�x

and k � k denotes the matrix norm.�(r) is the energy function for the GG model.

In the case of the non-stationary Gaussian model (12) = 2 (i.e. convexfunction) and �2x is spatially varying. The advantage of this model is the existenceof a closed form solution of (14) in form of adaptive Wiener or Lee �lters [18]. Inthe case of stationary GG model the general closed form solution does not exist,since the penalty function could be non-convex for < 1. In practice, iterativealgorithms are often used to solve this problem. Examples of such algorithmsare the stochastic [13] and deterministic annealing (mean-�eld annealing) ([20]),graduated nonconvexity [19], ARTUR algorithm [21] or its generalization [22].However, it should be noted that for the particular case of = 1, a closed formsolution in wavelet domain exists: it is known as soft-shrinkage ([23], [14]). Ofcourse, it is preferable to obtain the closed form solution for the analysis of theobtained estimate. To generalize the iterative approaches to the minimizationof the non-convex function (14) we propose to reformulate it as a reweightedleast squares (RLS) problem. Then equation (14) is reduced to the followingminimization problem:

x̂k+1 = argmin~x2RN f 1

2�2nky � ~xkk2 +wk+1krkk2g; (15)

where

wk+1 =1

rk�0(rk); (16)

rk =xk � �xk

�kx; (17)

�0(r) = [�( )] r

krk2� ; (18)

and k is the number of iterations. In this case, the penalty function is quadraticfor a �xed weighting fuction w.

Assuming w is constant for a particular iteration step, one can write thegeneral RLS solution in the next form:

x̂ =w�2n

w�2n + �2x�x+

�2xw�2n + �2x

y: (19)

This solution is similar to the closed form Wiener �lter solution [18]. The sameRLS solution could also be rewritten in the form of Lee �lter [18]:

x̂ = �x+�2x

w�2n + �2x(y � �x): (20)

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The principal di�erence with classical Wiener or Lee �lters is the presence ofthe weighting function w. This weighting function depends on the underlyingassumptions about the statistics of the cover image. In the rest of this paper,we will only consider the Lee version of the solution which coincides with theclassical case of Gaussian prior of the cover image (w = 1). It is important tonote that the shrinkage solution of image denoising problem previously used onlyin the wavelet domain can easily be obtained from (15) in the next closed form:

x̂ = �x+max(0; jy� �xj � T )sign(y � �x);

where T =�2n�x

p2 is the threshold for practically important case of Laplacian

image prior. This coincides with the soft-thresholding solution of the image de-noising problem [14].

The properties of the image denoising algorithm are de�ned by the multi-plicative term:

b :=�2x

w�2n + �2x(21)

in equations (19) and (20). It is commonly known, that the local variance is agood indicator of the local image activity, i.e. when it is small, the image is at,and a large enough variance indicates the presence of edges or highly texturedareas. Therefore, the function b determines the level of image smoothing. Forexample, for at regions �2x ! 0, and the estimated image equals local mean,while for edges or textured regions �2x >> �2n, b! 1 and the image is practicallyleft without any changes. Such a philosophy of the adaptive image �ltering is verywell matched with the texture masking property of the human visual system:the noise is more visible in at areas and less visible in regions with edges andtextures. Following this idea we propose to consider the texture masking propertyaccording to the function b based on the stochastic models (12) and (13).

7 Texture Masking Function

In order to be in the framework of the existing terminology, we propose to relatethe texture masking function to the noise visibility function (NVF) as:

NV F = 1� b =w�2n

w�2n + �2x; (22)

which is just the inverted version of the function b.In the main application of the proposed NVF in context of watermarking,

we assume that the noise (watermark) is an i.i.d. Gaussian process with unitvariance, i.e.N (0; 1). This noise excite the perceptual model (22), in analogy withthe AR image model. The NVF is the output of the perceptual model (22) to anoise N (0; 1). This model is schematically shown in �gure (2). The particularitiesof the developed perceptual model are determined by the weighting w, which arede�ned according to eqs. (17) and (18).

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7.1 NVF Based on Non-Stationary Gaussian Model

In the case of non-stationary Gaussian model the shape parameter is equal to2 and the autocovariance function is de�ned according to (10). The weightingfunction w is then equal to 1 according to equation (17) and NVF can simplybe written in the form:

NV F (i; j) =1

1 + �2x(i; j); (23)

where �2x(i; j) denotes the local variance of the image in a window centered onthe pixel with coordinates (i; j), 1 � i; j � M . Therefore, the NVF is inverselyproportional to the local image energy de�ned by the local variance. In order toestimate the local image variance the maximum likelihood (ML) estimate can beused. Assuming that the image is a locally i.i.d. Gaussian distributed randomvariable, the ML estimate is given by:

�2x(i; j) =1

(2L+ 1)2

LXk=�L

LXl=�L

(x(i + k; j + l)� �x(i; j))2 (24)

with

�x(i; j) =1

(2L + 1)2

LXk=�L

LXl=�L

x(i+ k; j + l); (25)

where a window of size (2L+1)�(2L+1) is used for the estimation. This estimateis often used in practice in many applications. However, the above estimate isassymptotically unbiased. To decrease the bias, it is necessary to enlarge thesampling space. From the other side, enlarging the window size violates therequirement of data being locally Gaussian, since the pixels from di�erent regionsoccur in the same local window. In order to have a more accurate model, itis reasonable to assume that at regions have a Gaussian distribution whiletextured areas and regions containing edges have some other highly-peaked,near-zero distribution (for example Laplacian). This assumption requires theanalysis of a mixture model with the Huber energy function (so called Huber-Markov RandomFields) and is the subject of our ongoing research. In this paper,we concentrate on the analysis of the eq. (22) and its relation to the stationaryGG version of NVF.

7.2 NVF Based on Stationary GG Model

For the stationary GG model we rewrite eq. (22) in the following form, takingeqs. (17) and (18) into account:

NV F (i; j) =w(i; j)

w(i; j) + �2x; (26)

where w(i; j) = [�( )] 1kr(i;j)k2�

and r(i; j) = x(i;j)��x(i;j)�x

.

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The particularities of this model are determined by the choice of two param-eters of the model, e.g. the shape parameter and the global image variance�2x. To estimate the shape parameter, we use a moment matching method asthe one used in [17]. The analysis consists of the next stages. First, the image isdecomposed according to the equation (9), using equation (25) as an estimateof the local mean. In the second stage, the moment matching method is appliedto the residual image and the shape parameter and the variance are estimated.Some typical examples of these estimations are shown in �gure (3). The shapeparameter for most of real images is in the range 0:3 � � 1. As a compari-son with frequently used pdfs such as Gaussian and Laplacian (also known asdouble-exponential), we have given the plots of these distributions and their cor-responding penalty (energy) function in �gures 4(a) and 4(b), respectively. Thederivatives of these energy functions and their corresponding weighting func-tions are shown in �gures 4(c) and 4(d). Comparing these pdfs, it is clear thatsmaller shape parameters lead to a distribution, which is more highly-peakednear zero. Another important conclusion regards the convexity of the energyfunction, which is concave (non-convex) for < 1, convex for = 1 and strictlyconvex for > 1. This causes the known properties of this non-convex functionin the interpolation applications and the corresponding discontinuity preserva-tion feature, when it is used in restoration or denoising applications. Despite ofthe nice edge preserving properties, the use of the non-convex functions causesthe known problems in the minimization problem (15). However, in the scope ofthis paper we are mostly interested in the analysis of the properties of the term(26), under condition that the cover image is available for designing the contentadaptive watermarking scheme.

8 Stochastical NVFs and Empirical Models

In the derivation of the content adaptive function we explicitly used stochasticmodelling of images. However, it is also very important to investigate the rel-evance of these analytical results with the empirically obtained results, whichre ect the particularities of the human visual system.

Obviously, the development of a complete model of the HVS that re ectsall its particularities is quite a di�cult task. Therefore, most of the developedempirical models utilize only the main features which are important for certainapplications, and give reasonable approximations for the practical use. Such sortof models have been used in deterministic image denoising and restoration algo-rithms ([24], [25]). They have also been used in the �eld of image compressionto reduce the visual artifacts of the lossy compression algorithms and to de-sign special quantization schemes (see for applications in DCT domain [26], andin wavelet domain [27] and [28]). This fact explains the appearance of a greatamount of watermarking algorithms based on the transform domain embedding([5]). Another reason that motivates the use of the transform domain is the de-crease of the inter-pixel redundancy by performing image decomposition in thetransform domain. In our case image decomposition is obtained in a natural way

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directly in the coordinate domain by decomposing the image into low-frequencyand high-frequency fractions. This is similar to the Gaussian and Laplacian pyra-mids and to the wavelet decomposition, where scale and orientation (3 directions)are additionally exploited.

To take the texture masking properties into account, it was proposed in [24]to use the NVF for image quantization in the coordinate domain in predictionschemes and also to extend it to the problem of image denoising. The most knownform of the empirical NVF is widely used in image restoration applications [25]:

NV F (i; j) =1

1 + ��2x(i; j); (27)

where � is a tuning parameter which must be chosen for every particular image.This version of NVF was the basic prototype for a lot of adaptive regularizationalgorithms.

Comparing the above function with the stochastically derived NVF basedon the non-stationary Gaussian model, it is very easy to establish the similaritybetween them. The only di�erence is the tuning parameter � which plays therole of the contrast adjustment in NVF. To make � image-dependent, it wasproposed to use:

� =D

�2xmax; (28)

where �2xmax is the maximum local variance for a given image and D 2 [50; 100]is an experimentally determined parameter.

9 Content Adaptive Watermark Embedding

Using the proposed content adaptive strategy, we can now formulate the �nalembedding equation:

y = x+ (1� NV F ) � n � S; (29)

where S denotes the watermark strength. The above rule embeds the watermarkin highly textured areas and areas containing edges stronger than in the the atregions.In very at regions, where NVF approaches 1, the strength of the embeddedwatermark approaches zero. As a consequence of this embedding rule, the water-mark information is (nearly) lost in these areas. Therefore, to avoid this problem,we propose to modify the above rule, and to increase the watermark strength inthese areas to a level below the visibility threshold:

y = x+ (1� NV F ) � n � S +NV F � n � S1; (30)

with S1 being about 3 for most of real world and computer generated images.We are now investigating more complex cases, to replace this �xed value by animage-dependent variable, which takes the luminance sensitivity of HVS intoaccount.

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Finally, the watermark detection and the message demodulation will be ac-complished in accordance with the general scheme presented in (1), where theestimation x̂ of the cover image is done using the image denoising algorithm(15).

The performance of the resulting watermarking algorithm depends stronglyon the particular scheme designed to resist geometrical distortions or the basicimage processing operations such as �ltering, compression and so on. Exam-ples of such distortions are integrated in StirMark watermarking benchmark [2].For instance, the coordinate domain method given in[1] uses properties of theautocorrelation function of a spatially spreaded watermark, while the Fourierdomain approach developed by [3] utilizes a pre-de�ned template to detect andcompensate the undergone geometrical distortions. The adaptive approach pro-posed here can be integrated into these methods to achieve the best trade-o�between the two contradicting goals of increasing the robustness by increasingthe watermark strength and at the same time, decreasing the visual artifactsintroduced by the watermarking process. This is the subject of our current re-search.Here, we address mainly the problem of content adaptive watermark embedding,and investigate its visibility aspects in the next section.

10 Results of Computer Simulation

To illustrate the main features of the proposed content adaptive embeddingmethod, we tested our algorithm on a number of real world and computer gen-erated images. In this paper we will restrict our consideration to several imageswith most typical properties. The �rst stage of modeling consists of the calcu-lation of the NVFs according to the developed stochastic models of the coverimage. The NVFs based on non-stationary Gaussian (a) and stationary GG (b)models for Barbara and Fish images are shown in Figures (5) and (6), respec-tively. The NVF calculated according to the non-stationary Gaussian model issmoother and the intensity for the edges is about the same order as that of tex-tured regions. The NVF calculated from stationary GG model looks more noisydue to the discontinuity preserving properties of the non-convex energy functionof GG prior. From the point of view of noise visibility, both functions re ectthe regions of reduced visibility. This corresponds to the desirable features of acontent adaptive watermarking approach. To demonstrate the possibility of inte-grating this adaptive approach into transform domain based techniques, anotherset of experiments was performed, which demonstrate the relationship betweencoordinate domain and Fourier domain properties of the cover image and thecorresponding NVFs. The magnitude spectrum of Camaraman image (7a) andthe corresponding spectra of the NVFs based on the non-stationary Gaussian(7b) and on the stationary GG (7c) model were calculated. It is important tonote that the spectra of real images are characterized by a high level of spa-tial anisotropy, where the principal component directions are determined by theproperties of the image. These directions in the spatial spectrum of the original

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image are determined by the spatial orientation of the edges and textures in thecoordinate domain and play a very important role for the high quality imageaqcuisition systems under resolution constraints of imaging aperture [30] . Theprospective watermarking techniques have to take these properties into account,to survive denoising or lossy compression attacks. Such a strategy implies thatthe watermark should be most strongly embedded in these directions. The com-parison of the spectrum of the cover image with the spectra of correspondingNVFs, shows that the directions of the principal components coincide. This canbe the solution to �nding a compromise between making the watermark as ro-bust as possible and avoiding the visible distortions that are introduced by thewatermarking process. The next stage of modeling is the comparison of the vi-sual quality of the stego images generated using the di�erent developed NVFs.The test consists in the direct embedding of a Gaussian distributed watermarkaccording to the equation (1) and to the content adaptive schemes described inequation (30), using both NVFs. The strength of the watermark (i.e. its stan-dard deviation) is equal to 5, 10 and 15 for the non-adaptive scheme (1). Thecorresponding results are shown in Figure (8) in parts (a),(d) and (g). The peak-signal-to noise ratio (PSNR) was chosen as the criterion for the comparison ofthe introduced distortions in the stego image or equivalentlly to estimate thegeneral energy of the watermark

PSNR = 10 log10k255k2kx� yk2

The resulted PSNRs for Barbara image are gathered in table (1).

Embedding method Barbara: Watermark strength

Non-adaptive 5:0 10:0 15:0Adaptive non-stationary Gaussian NVF 6:5 15:0 23:0

Adaptive stationary GG NVF 8:0 20:0 31:0Adaptive Kankanhalli 9:8 19:6 30:38

PSNR (dB) 34:1 28:3 24:7

Table 1. Comparison of the watermark strength for di�erent watermark embeddingmethods

To receive similar PSNR values in case of adaptive embedding, the water-mark strength was increased in these cases to the values given in table (1). Theresulting images are depicted in �gure (8), parts (b), (e), (h) and �gure (8),parts (e), (f), (i) for the non-stationary Gaussian and the stationary GG basedNVFs, respectively. The corresponding images in the appendix show that eventhe PSNR is very similar, the image qualities are quite di�erent. Comparingboth visual quality and objective parameters given by the watermark strength,

14

Embedding method Fish: Watermark strength

Non-adaptive 5:0 10:0 15:0Adaptive non-stationary Gaussian NVF 6:5 14:5 23:0

Adaptive stationary GG NVF 9:0 21:0 35:0Adaptive Kankanhalli 10:5 21:7 32:2

PSNR (dB) 34:64 28:67 25:26

Table 2. Comparison of the watermark strength for di�erent watermark embeddingmethods

the adaptive scheme based on NVF calculated from stationary GG model seemsto be superior to other schemes. This can be explained regarding the abovementioned properties of the local variance estimation in equation (23), given bythe ML-estimate (24). In particular, the watermark is very visible on the edges,because here the requirement of data being local stationary is violated. Thisdecreases the performance of the ML-estimate. In the stationary GG model,only the global variance estimation is needed (equation (26)), where the data isconsidered to be globally i.i.d.. Therefore, distortions due to edges are consider-ably smaller than in the non-stationary Gaussian model embedding scheme. Thesame experiments were performed for Fish image and the results are shown intable (2). In some watermarking algorithms [1], the watermark is embedded inthe coordinate domain as a pseudo-random spatially spreaded binary sequence-1;1. To investigate the visibility aspects of this watermark in the scope of theproposed approach, a similar test was performed. The results are shown in �gure(9). The PSNR values and the watermark strengths practically coincide with thedata given in table (1).

Based on the empirical evaluation of di�erent test people, we think that theimage quality of our approach is better than the results of the Kankanhalli ap-proach (see the corresponding stego images in the appendix(�gures 11, 12). Thestationary GG model shows a superior performance in this case too. The thirdstage of the modeling consists in the investigation of the watermark robustnessagainst adaptive Wiener denoising. The resulting PSNR of the watermark afterthe above attack was determined as

PSNRn = 10 log10k255k2

kn� nak2

where na is the watermark after attack. The results of this experiment are pre-sented in table (3). The watermarks before and after this attack are shown in�gure (10) for a PSNR equal to 28:3 dB. Inspection of the results of �gure (10)show that the watermark survives mostly in the highly textured regions andareas with edges, where its strength was increased according to the developed

15

embedding method watermark PSNR (dB)(after adaptive Wiener denoising)

non-adaptive 25:99adaptive non-stationary Gaussian NVF 26:86

adaptive stationary GG NVF 26:97

Table 3. The watermark PSNR after adaptive Wiener denoising attack for di�erentwatermark embedding methods

content adaptive scheme. Therefore, the obtained results clearly indicate theadvantages of the proposed approach for digital image watermarking.

11 Conclusions and Future Work

In this paper we have presented a new approach for content adaptive water-marking which can be applied to di�erent watermarking algorithms based onthe coordinate domain, the Fourier or wavelet domains. In contrast to state-of-the-art heuristic techniques our approach is based on a stochastic modellingframework which allows us to derive the corresponding analytic formula in aclosed form solution. It is, therefore, not necessary to investigate in practicethe di�erent parameter sets for di�erent image classes but to apply mathemat-ical expressions which can be easily computed. In addition, we have shown theclose relationship of the derived optimization problems and the correspondingsolutions to image denoising techniques. We have shown that some of the heuris-tically derived solutions are special cases of our model. Running di�erent testswe have sucessfully compared our approach against other techniques. The testresults have also shown that some assumptions in many information theoreticpapers for watermarking are not satis�ed, namely, the distribution of the im-age is not stationary Gaussian and the channel capacity is not uniform meaningthat depending on the sample, the image contents constitute a channel capacitywhich is closely linked to the local image characteristics which are not uniform.It is, therefore, necessary to consider the enhanced information theory based ap-proaches which satisfy these constraints. Based on the identi�ed relationship toimage denoising problems, we are going to develop the corresponding attacks oncommercial watermark schemes to test our models as the basis of a new piracytool and for further enhancement of the existing watermarking techniques. Inthe future, we will apply the presented model also for the construction of morerobust encoder and decoder schemes.

12 Acknowledgements

We appreciate usefull comments of S. Pereira, G. Csurka and F. Deguillaumeduring the work on the paper. S. Voloshynovskiy is grateful to I. Kozintsev, K.

16

Mihcak, A. Lanterman and Profs. P. Moulin and Y. Bresler for fruitfull and help-full discussions during his staying at University of Illinois at Urbana-Champaign.

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13 Appendix

Estimatory n Robust detector

and demodulator

Messagex

Key

Fig. 1. Block diagram of the watermark detection algorithm according to equation (2).

Perceptual

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NVFn~N(0,1)

Fig. 2. Generation of the NVF from perceptual model.

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Fig. 11. The Barbara stego images calculated for the Kankanhalli (a) and the station-ary GG model (b).

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Fig. 12. The Fish stego images calculated for the Kankanhalli (a) and the stationaryGG model (b).